The classical Kermack-McKendrick homogeneous SIR （susceptible, infected and removed） model is well known, Its general solution is a function of the unique parameter （the reproduction number） that is equal to a mea...The classical Kermack-McKendrick homogeneous SIR （susceptible, infected and removed） model is well known, Its general solution is a function of the unique parameter （the reproduction number） that is equal to a mean number of secondary cases produced by a typical infected individual in a completely susceptible population. If the reproduction number is more than one （the threshold value） its value describes an epidemic scope： larger values correspond to more severe epidemics. In the more complex compartment SIR models the population is divided into several non-overlapping groups. It allows us to partly remove assumptions of the classical model. It is well known that for this kind of models, just as for the classical model there is the threshold parameter R0. Usually it is called by the same name--the reproduction number--though the physical meaning of this parameter has changed. The main purpose of the paper is to show that this new parameter is a not unique measure of an epidemic severity for any compartment SIR model. In particular it means that for such models comparison of the severity of two epidemics by simple comparing values of their reproduction numbers is incorrect. For compartment models these statements were proved with the help of the corresponding ODEs analysis. Very popular now individual-based models （IBMs） are more complex in comparison with the compartment ones since they use overlapping groups （school children are members of families also, for example）. In such a case Diekmann＇s calculation method for the reproduction number used in many papers is inapplicable as well as a presentation the simulation results obtained as functions of this parameter.展开更多
In this paper, we consider the backward Euler discretization derived from a continuous SIRS epidemic model, which contains a remaining problem that our discrete model has two solutions for infected population; one is ...In this paper, we consider the backward Euler discretization derived from a continuous SIRS epidemic model, which contains a remaining problem that our discrete model has two solutions for infected population; one is positive and the other is negative. Under an additional positiveness condition on infected population, we show that the backward Euler discretization is one of simple discrete-time analogue which preserves the global asymptotic stability of equilibria of the corresponding continuous model.展开更多
基金Acknowledgements This work was assisted through participation in ＂Optimal Control and Optimization for Individual- based and Agent-based Models＂ Investigative Workshop at the National Institute for Mathematical and Biological Synthesis, sponsored by the National Science Foundation, the U.S. Department of Homeland Security, and the U.S. Department of Agriculture through NSF Award #EF-0832858, with additional support from The University of Tennessee, Knoxville.
文摘The classical Kermack-McKendrick homogeneous SIR （susceptible, infected and removed） model is well known, Its general solution is a function of the unique parameter （the reproduction number） that is equal to a mean number of secondary cases produced by a typical infected individual in a completely susceptible population. If the reproduction number is more than one （the threshold value） its value describes an epidemic scope： larger values correspond to more severe epidemics. In the more complex compartment SIR models the population is divided into several non-overlapping groups. It allows us to partly remove assumptions of the classical model. It is well known that for this kind of models, just as for the classical model there is the threshold parameter R0. Usually it is called by the same name--the reproduction number--though the physical meaning of this parameter has changed. The main purpose of the paper is to show that this new parameter is a not unique measure of an epidemic severity for any compartment SIR model. In particular it means that for such models comparison of the severity of two epidemics by simple comparing values of their reproduction numbers is incorrect. For compartment models these statements were proved with the help of the corresponding ODEs analysis. Very popular now individual-based models （IBMs） are more complex in comparison with the compartment ones since they use overlapping groups （school children are members of families also, for example）. In such a case Diekmann＇s calculation method for the reproduction number used in many papers is inapplicable as well as a presentation the simulation results obtained as functions of this parameter.
文摘In this paper, we consider the backward Euler discretization derived from a continuous SIRS epidemic model, which contains a remaining problem that our discrete model has two solutions for infected population; one is positive and the other is negative. Under an additional positiveness condition on infected population, we show that the backward Euler discretization is one of simple discrete-time analogue which preserves the global asymptotic stability of equilibria of the corresponding continuous model.
